Antiquantum $q$-series identities and mock theta functions

TL;DR

This paper extends the theory of mock theta functions by establishing antiquantum $q$-series identities for Ramanujan's third order mock theta functions, linking boundary behaviors at roots of unity with modular eta-quotients.

Contribution

It introduces and proves antiquantum $q$-series identities for all third order mock theta functions, expanding the understanding of their boundary behaviors and connections to quantum modular forms.

Findings

Established antiquantum $q$-series identities for third order mock theta functions.

Connected boundary behaviors at roots of unity with modular eta-quotients.

Developed more general identities of independent mathematical interest.

Abstract

Ramanujan's original definition of mock theta functions from 1920 involves their asymptotic behaviors at roots of unity on the boundary of the disk of convergence $|q|<1$. More recently this topic has been related by several authors, including the first author with Ono and Rhoades in 2013, to quantum modular forms, first defined in 2010 by Zagier. In 2021, Lovejoy defined and studied related quantum $q$-series identities, which do not hold as equalities between power series inside the disk $|q|<1$ but which do hold on dense subsets of roots of unity on the boundary. Inspired by this, in our prior joint work from 2024 we further studied quantum $q$-series identities as related to mock theta functions and quantum modular forms; we also defined and studied antiquantum $q$-series identities, between series which are equal inside the disk $|q|<1$ but which hold at dense sets of roots of unity on the boundary for which one of the series diverges and is unnaturally truncated. Here, building from our previous work, we establish antiquantum $q$-series identities for all of Ramanujan's third order mock theta functions. We deduce these results in part by establishing and applying more general identities which are also of independent interest, and by using the theory of modular eta-quotients.